Prove that $\displaystyle\prod_{q\in \mathbb{Q}^{\times}}|q|=1$ Prove that $\displaystyle\prod_{q\in \mathbb{Q}^{\times}}|q|=1$. I don't have a lot of experience working with infinite products, but I read a couple of theorems that say that absolute convergence of infinite products requires that $\prod1+|a_n|$ converges, and that $\prod1+|a_n|$ converges iff $\sum a_n$ converges. Now $\sum_{q\in \mathbb{Q}^{\times}}|q|$ certainly does _not_ converge, implying that my original product is not absolutely convergent. Which leaves me with the problem of being unable to rearrange its terms. But since I was never given an enumeration of my rationals to begin with, I'm a bit vexed as to how I should proceed. Here is the link to the problem: homework 2 (problem 2). I'm not in the class, just doing the homeworks. I'm doing it for the $\mid \cdot \mid_{\infty}$ absolute value. Which is supposed to be just the normal absolute value (according to homework 1).

You've misstated the problem. For all $x\in\mathbb{Q}^\times$, $$\prod_{p\text{ prime or }\infty}|x|_p$$ is an infinite product all but finitely many terms of which are 1, so it certainly converges.

**Hint:** Note that, if $x=p_1^{a_1}\cdots p_n^{a_n}$ where the $p_i$ are primes and $a_i\in\mathbb{Z}$, then $$|x|_{p_i}=\frac{1}{p_i^{a_i}}$$ and that $|x|_p=1$ for all other primes.